The Mathematics of Music (MAT 1631)

Instructor: Dr. Rachel Hall
Office: 229 Barbelin
Office Hours: T 10:30-11:30, W 9:30-10:30, F 12-1
Telephone: (610) 660-3096 (Office)

Course Description: Music has many connections to mathematics.  The ancient Greeks discovered that chords with a pleasing sound are related to simple ratios of integers.  The mathematics of rhythm has also been studied for centuries—in fact, ancient Indian writers discovered the Fibonacci sequence in the rhythms of Sanskrit poetry.  Other connections between math and music include the equations describing the sounds of musical instruments, the mathematics behind digital recording, the use of symmetry and group theory in composition, the exploration of patterns by African and Indian drummers, the application of chaos theory to modeling the behavior of melodies, and the representation of chords by exotic geometric objects called orbifolds.  Along the way, we discuss the role of creativity in mathematics and the ways in which mathematics has inspired musicians.


Course Objectives:  This course introduces a number of mathematical topics and investigates their applications to the analysis of music.  I intend to use the medium of musical analysis to (1) explore mathematical concepts such as Fourier series and tilings that are not covered in other math courses, and (2) introduce topics such as group theory and combinatorics covered in more detail in upper-level math courses.  The course is not proofs-based.  Students will complete a semester-long project that explores one aspect of the course in depth.


Prerequisite:  Calculus II and some musical training (Music 1511 or the equivalent).  Students with exceptional performance in Calculus I (or AP) and musical training will be admitted on a case-by-case basis. 


Texts: Daniel Levitin, This is Your Brain on Music, Dutton, 2006.  David Benson, Music: a Mathematical Offering, Cambridge University Press, 2006.  This text is available free online or for purchase at the bookstore. (Warning:  the page numbers in the online version are different from the printed one.  I will refer to the numbers in the printed book in class.)

Quizzes:  There will be a 15-minute quiz given in class every Wednesday when there is not a problem set due, giving ten quizzes.  Quizzes are based on homework “drill” problems, readings, and class notes.  There are no makeup quizzes, but your lowest two grades will be dropped.

Problem sets: There will be four problem sets, due February 6, February 29, March 28, and April 18 (all Fridays).  Strict academic honesty policies apply.


Project:  Projects explore one or more aspects of the course in depth.  The final project includes three presentations and a written paper plus supporting materials as appropriate.  Anyone who submits a project that I judge to be of publishable quality and submits it to a journal or competition will receive an A for the course.

Grading scale:  Grades are calculated out of 500 points:

Four problem sets

50 points each

Final project

200 points

Ten quizzes (lowest two dropped)

10 points each

Class preparation and participation

20 points

The cutoffs are 466 A, 450 A-, 433 B+, 416 B, 400 B-, 366 C, 350 C-, 333 D+, 300 D, and below 300 F.  Grades may be curved at the end of the semester.

Academic Honesty: Dishonesty includes cheating on a test, falsifying data, misrepresenting the work of others as your own (plagiarism), and helping another student cheat or plagiarize. At the very least, an academic honesty infraction will result in the filing of a violation report and a grade of zero on that particular assignment; serious or repeated infractions of the Academic Honesty policy will result in failure of the course. For complete information about the University’s policy on Academic Honesty, consult the Student Handbook 2007-2008.

Attendance: Class attendance is mandatory.  Although I do not have a rigid cut policy, anyone who has missed lots of classes and is doing poorly in the course should not expect much sympathy from me.  If you do miss a class, it is your responsibility to obtain the notes and assignments from another.  (There are no makeup quizzes.)

Students with Disabilities:  Policies for students with disabilities are posted on the course web site.


Schedule (subject to change):




Poetic meter

Intro to combinatorics; recursion and iteration; de Bruijn sequences

Analysis of drumming patterns in world music

Phylogenetic analysis of binary sequences; asymmetric rhythms; tiling canons

Review of music notation


Problem set 1, due February 6

The physics of musical instruments, tuning

Wave equation and intro to PDE’s, Fourier series, beats

Digital recording and synthesis

Intro to signal processing:  sampling, DFT, aliasing

Project proposal presentations (date TBA)

Problem set 2, due February 29

Scale theory

Melodic transformations (inversion, retrograde, etc)

Harmonic transformations (transposition, inversions)

12-tone theory

Neo-Riemannian set theory

Equivalence relations; group theory;

Burnside’s lemma

Project progress report presentations (date TBA)

Problem set 3, due March 28

Harmony, counterpoint, voice leading

Tilings, quotient spaces (orbifolds), elementary topology

Problem set 4, due April 18

(Optional) Melodic models

1/f noise

Project consultations

Final project presentations

Friday, May 2, 11:30 am - 1:30 pm